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Mathematics

Mathematics

The following results in the branches Euclidean geometry, set theory, linear programming, nonstandard analysis, topology and number theory have to be called sensational! Known statements and elementary concepts such as axiom, field, etc., are as given in the relevant literature or on Wikipedia. They are only given when they deviate from the conventional usage or when clarification is necessary (between multiple options).

Unlike conventional usage in which brackets denote a more detailed explanation, bracketed parts of a statement can either be included or excluded: the statement is valid in both cases. The end of a proof is indicated by the symbol \(\square\). Because of the finiteness of our world, there are certain difficulties to treat the infinite. If numbers are zeroes of a polynomial having rational coefficients, whose largest exponent of its argument is its degree, they are called algebraic, otherwise transcendental.

We may ask whether we define the number of elements of the set of algebraic numbers finite or infinite. From algebra, we know that the sum, difference, product, and quotient of two algebraic numbers of natural degree \(m\) or \(n\) are algebraic of degree at most \(mn\), and that the \(1/m\)-th power of an algebraic number of degree \(n\) is also algebraic of degree at most \(mn\). Transcendental numbers can be viewed as the sum of an algebraic part and a transcendental remainder.

When investigating whether a number is transcendental, if the remainder may be expressed as the limit value of a zero sequence \(\left({a}_{n}\right)\), we may not simply disregard the values of the sequence for large \(n\): They are important. Transcendental numbers are the numbers that lie between algebraic numbers or on either side of them. Set theory shows, if two distinct transcendental numbers (algebraic of degree \(m\)) are sufficiently close, that there is no algebraic number \(< m\)) between them.

Together with the finite rational numbers, the infinite (complex-)rational numbers already numerically make up the entirety of all real (complex) numbers. Therefore, algebraic and transcendental numbers are (numerically) difficult to distinguish, and approximations are of little use when determining whether a number is algebraic (to a certain degree). Real continued fractions that do not terminate as a rational number are transcendental, since they are infinite rational. They can only be used as an approximation of algebraic numbers.

Since all real numbers are (approximately) (infinite) rational numbers, we can compute them in real-time. In the case of transcendental numbers we must satisfy ourselves with an arbitrarily precise infinitesimal number, unlike in the case of algebraic numbers, where we can argue using the corresponding minimal polynomial or series. We can therefore represent them in the form of a rational quotient with infinite numerator and denominator.

The finite definition offers significant advantages of handling and is traditional. It is shown that the sets of natural, integer, rational, algebraic, real or complex numbers are not closed. Hence, the difference of algebraic numbers is no longer necessarily algebraic, what complicates the theory of transcendental numbers. An infinite rational and transcendental number can also consist of a finite continued fraction.

Here the last partial denominator can be infinitely big. If we would identify it with a conventionally rational number by setting the last partial fraction equal to zero, it would simultaneously solve a linear equation with (infinite) integer coefficients. This identification leads to contradictions, if the first equation is subtracted several times from the second one and the solution of every newly emerged equation is determined.

It is correct to work with approximate fractions. We can show that, by (conventionally natural and infinite natural) induction, starting with the set of conventionally natural numbers, it can be diagonalised up to any power according to Cantor, so that all(!) infinite sets are equipotent to the set of conventionally natural numbers if we use Hilbert's translations as an aid.

This contradictoriness is met with the theorem that there is for no set a bijection to its proper subset. Hence, Dedekind-infinity and Hilbert's hotel are refuted, since the image sets of translations of every set lead out of the latter. We can specify every number of elements of a set by reference to the set of conventionally natural numbers. We obtain this only by precisely prescribing the exact construction.

It is something completely different whether we consider all multiples of five and thereby construct the associated set in such a way that each conventionally natural number is multiplied by five, or whether we remove all numbers, up to each fifth, from the set of the conventionally natural numbers. Cantor regarded such sets as of the same cardinal number. But if we consider bijections correctly, we get a different picture. The fuss made so far over ordinal and cardinal numbers is omitted.

Cantor's distinction between merely countable and uncountable sets is too undifferentiated. The correct treatment of bijections yields the statement that, concerning the number, there are infinite many sets between the set of conventionally natural numbers and the one of conventionally real numbers. Thus, the continuum hypothesis gets a new answer. Furthermore, the asymptotic function of the number of algebraic numbers is determined.

The set \(\mathbb{R}\) aof all real numbers is isomorphic to a set of (hyper-) natural or integer numbers. It has both a fixed minimum element and a fixed maximum element, since we view \(\mathbb{R}\) holistically and completely, from which we have together with the field axioms its closure. Otherwise, we maybe would have to adapt our theory again and again to the circumstances.

Admittedly, it follows some dialectics from the conventional irrationality proof of \(\sqrt{2}\) and the multiplicative inverses are usually only approximately contained in \(\mathbb{R}\). Many of the conclusions derived here can be extended to sets with other upper and lower bounds and more generally to metric spaces. We will not list the results for which this is possible, since the associated arguments are easy.

A disk without its boundary conventionally represents an open set, because then each point of it has a conventional neighbourhood that lies completely in this set. Here, the idea is based on that, if the points are considered on a half-line, starting at the centre of the disk, it must always be considered a (real) neighbourhood for each point on this half-line towards the boundary.

In fact, however, "the end of the flagpole" must sometime be reached. So there must be a point in the interior of the disk, which has no conventional neighbourhood in this interior. Therefore, the term openness for sets is inept. If the unit disk is considered around the origin of ordinates, so the last point of the half-line \([0, 1[\), dually represented, is the point \(0,\overline{1}_{2}\) and the next point is the boundary point 1.

Between these two points lies no other point. The former has no neighbourhood that lies in the interior of the disk, though it is an interior point. For this reason, the disk without boundary is also closed, since the last points of the half-line form, beginning from the centre of the disk, just the closure as boundary. Since the neighbourhoods do not exist on their boundary, the term closure is meaningless for sets in Euclidean space.

Therefore it holds that at least in Euclidean space every open set is also closed, what reduces these terms to absurdity. This has not been further annoying, since infinitesimal quantities so far were not considered in a differentiated manner, thus in particular the numbers 1 and \(0,\overline{1}_{2}\) were equated. However, this is incorrect as is explained in Set Theory, since otherwise algebraicity (1) and transcendence \((0.\overline{1}_{2})\) are equated.

The absurd can also be illustrated by the fact that an infinite intersection of open sets such as all open concentric disks can form a closed set (the common centre of the disk). An infinite union of closed sets can build an open set as an open disk does, as a union of all of its points as closed sets.

A 0-dimensional set (point) is therefore open, because every neighbourhood, also consists of one point. Hence, the empty set \(\emptyset\) is also closed, and as a consequence the whole Euclidean space is closed. Using spheres, that what has been said can easily be generalised to higher dimensions. The terms of the inner and outer point remain meaningful however, if any infinitesimal radiuses are permitted.

Since an absurd and meaningless special case also makes the general case here absurd or meaningless, one also for metric and topological spaces openness or closure of sets should not be considered, particularly since the definition of a conventional topological space appears oddly content-free and arbitrary, while the terms of interior and exterior point as well as boundary point are still useful and appropriate.

The Riemann series theorem is false, since when summing positive summands to a desired value we are forced to add negative values until the original sum is attained, and vice versa. The same is true in the case that we obtain a smaller or larger value than the sum of the positive or negative terms, since the remainders almost fully cancel, and so on. We must avoid choosing arbitrary methods to deal with infinities if we wish to avoid making mistakes.

We may not neglect the existence of something because it is stored at infinity. The definiton of the exact integral in nonstandard analysis according to a rectangular rule maybe requires error estimations (see the literature on Numerical Mathematics). It should be noted that neither the integral nor the derivative is assumed to be continuous. An alternative definition of the exact volume integral may be given as described further below. However, the original definitions are easier to manipulate.

In some cases, suitable Landau notation may be useful. If the result of differentiation lies outside of the domain, it should be replaced by the closest number within the domain. If this is not uniquely determined, the result can either be given as the set of all such numbers, or we can select the preferred result (e.g. according to a uniform rule). Actual integration (as the inverse operation to differentiation) only makes sense for continuous functions if we wish to go beyond simple summation.

However, if we can express the function values in the form of finitely many continuous functions whose antiderivatives may be computed in finite time, integrals may be calculated even for discontinuous functions, if necessary by appropriately applying the Euler-Maclaurin summation formula and other simplification techniques. The exact integral is more general than Riemann, Lebesgue(-Stieltjes) integral and other types of integral.

The latter exist only in conventionally measurable sets, and function values may skip values, which are not measured then. Five examples given in nonstandard analysis illustrate its superiority and the strength of using infinitesimal and infinite values. The examples of functions are only for purposes of illustration so simple. In particular, we will consider inconcrete and infinite sets that are conventionally non-measurable, and functions that are conventionally discontinuous.

The definition of real numbers via Dedekind cuts is therefore just as unsuitable as its definition via equivalence classes of rational Cauchy sequences. Conventional differentiation and integration lose the ability to distinguish between transcendental and algebraic in the conventional process of taking limits. This is problematic, for example when we wish to exactly determine the roots of a polynomial. Therefore, we cannot preserve the conventional analysis in its existing form and proceed differently.

Since individual n-ness belongs to every natural number n that cannot be derived from its predecessors or successors, there is no complete system of axioms in mathematics, because with each new number something irreducible new emerges. By confining, however, to selected aspects, we can specify a finite system of axioms for a finite number of entities. Each level of infinity refuses completeness all the more.

Mathematics is not value-free. Theories are based on presuppositions. In mathematics, they are often expressed by axioms (prove to be true or false resp. to justify). Thus, all theories are incomplete and, as the case may be, beyond that, contradictory. Instead of explicit axioms, (implicit) definitions are more suitable in that the existence of the specified is tacitly presupposed with all emerging difficulties, until refutation.

The Euclidean geometry gives three definitions that challenge several axioms, using results of set theory. The question of a fair distribution of persons deals less about the theoretical content than about practical application. Here a spread-sheet analysis is used. The set theory defines some new sets and states generally their number, especially for the algebraic numbers.

The nonstandard analysis redefines integration, differentiation, continuity, convergence and limit value for finite and infinite (conventionally not measurable) sets (maybe by considering their homogeneity) and gives examples, as well as for discontinuous functions, to obtain better, more precise, more elegant or simply correct mathematical statements (also for known theorems of analysis). It solves the measure problem.

The advice is the result of my mathematical experience and discusses appropriate ways and procedures for solving mathematical problems. The topology redefines the (simple) connectedness of sets and states the real and complex numbers as spaces only with Fréchet topology resp. without Hausdorff property.

The linear programming proves first the diameter theorem for polytopes. The known exponential simplex method with a new perturbation method is confronted with the polynomial intex method, which also solves linear systems. On the internet, a linear programme can be entered as a table in one area, which is solved in a second area. A second solution will be returned if it exists.

In number theory, necessary and sufficient criteria for transcendental numbers are stated with some new examples. The conjecture by Littlewood is refuted and conventionally proven. The conjecture by Alaoglu and Erdős as well as the generalised Riemann hypothesis are elementarily proven. The latter implies a more precise prime number theorem and the ternary Goldbach theorem, as well as numerous further theorems.

In the computation of time, it is shown how the octal system can be used practically worldwide and what the advantages are here. Besides introducing a new calendar and a new calculation of clock time, the octal system is applied also to the SI-units metre (m) and second (s). It is reasoned with practical examples why the octal system can completely replace the decimal system advantageously.

© 2002-2019 by Boris Haase


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